Solving the bicriterion routing and scheduling problem for hazardous materials distribution

Transportation Research Part C 18 (2010) 713–726

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Transportation Research Part C journal homepage: www.elsevier.com/locate/trc

Solving the bicriterion routing and scheduling problem for hazardous materials distribution Konstantinos N. Androutsopoulos, Konstantinos G. Zografos * Transportation Systems and Logistics Laboratory, Department of Management Science and Technology, Athens University of Economics and Business, Evelpidon 47A and 33, Lefkados 113 62, Athens, Greece

a r t i c l e

i n f o

Article history: Received 7 January 2009 Received in revised form 14 October 2009 Accepted 5 December 2009

Keywords: Routing Dynamic programming Multiple criteria analysis Hazardous materials transportation k-Shortest path

a b s t r a c t Hazardous materials routing and scheduling decisions involve the determination of the minimum cost and/or risk routes for servicing the demand of a given set of customers. This paper addresses the bicriterion routing and scheduling problem arising in hazardous materials distribution planning. Under the assumption that the cost and risk attributes of each arc of the underlying transportation network are time-dependent, the proposed routing and scheduling problem pertains to the determination of the non-dominated time-dependent paths for servicing a given and fixed sequence of customers (intermediate stops) within specified time windows. Due to the heavy computational burden for solving this bicriterion problem, an alternative algorithm is proposed that determines the k-shortest time-dependent paths. Moreover an algorithm is provided for solving the bicriterion problem. The proximity of the solutions of the k-shortest time-dependent path problem with the non-dominated solutions is assessed on a set of problems developed by the authors. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction It is widely recognized that hazardous materials transportation risk may be drastically reduced by planning routes that minimise the accident probability and/or the expected consequences of an accident (List et al., 1991). Significant research has been focused on developing mathematical models and algorithms for determining safe and economical hazardous materials routes. The major part of the relevant research has been focused on routing decisions for transporting shipments of hazardous materials from an origin to a single destination minimising cost and/or risk related criteria. Recently, an alternative direction of research has arisen that is focused on the problem of designing the underlying hazardous materials transportation network (from the perspective of the public authorities) in order to minimise the network-wide risk. This approach is applicable to countries or regions (e.g. Alberta Canada) where the public authorities are entitled to exercise traffic control measures for banning hazardous materials shipments passing through specific parts of the roadway network where various sensitive facilities are located. An extensive literature review for the above type of problems is provided in (Erkut et al., 2007). Although substantial research effort has been focused on routing decision for the full truckload hazardous materials transportation, limited research has been placed on planning routes for the less-than-truckload hazardous materials transportation. A hazardous materials distribution route involves the service of a set of intermediate stops (customers) usually located in urban or suburban areas. In general, planning truck routes for freight distribution includes the following categories of decisions: (i) the determination of the sequence of stops serviced by each truck and (ii) the specification of the path followed by each truck (route-path) on the underlying roadway network in order to deliver the relevant orders. The latter deci-

* Corresponding author. Tel.: +30 210 82036735; fax: +30 210 8203684. E-mail address: [email protected] (K.G. Zografos). 0968-090X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.trc.2009.12.002

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sion is closely dependent on the former, since for each given sequence of stops the optimum path traversing the entire sequence of stops is envisaged. However, in the case of hazardous materials deliveries, determining the sequence of stops per truck usually involves various case-specific constraints (Caramia et al., 2007; Avella et al., 2004; Ronen, 1995; Brown et al., 1987; Brown and Graves, 1981). For instance, in the problem of delivering orders for multiple liquefied petroleum products (e.g. various types of gasoline and diesel) which cannot be mixed, the loading constraints constitute a significant criterion for specifying the sequence of visiting the customers (Brown et al., 1987). In this context, the problem of sequencing the stops serviced by a truck depends on the specific substances included in the corresponding orders and the associated (regulatory or operational) loading constraints, thus making it difficult to develop a model for the general case. However, assuming that the sequence of stops for a truck is known in advance, then the routing and scheduling problem reduces to a path-finding problem from an origin to a destination (not necessarily different from the origin) with a set of fixed mandatory intermediate stops. The focus of this paper in on analysing and solving the emerging path-finding problem for a given hazardous materials route. A major feature of the emerging hazardous materials path-finding problem, is that it involves multiple time-dependent routing criteria including the transportation cost expressed by the travel time and the transportation risk. In practice, a significant part of the underlying roadway network covers urban areas, where the travel speeds on the arcs of the network are assumed dynamic (Miller-Hooks and Mahmassani, 2000; Pattanamekar et al., 2003). Furthermore, the dynamic traffic control measures applied by the public authorities for diverting hazardous materials shipments from sensitive facilities (hospitals, schools, shopping centres), imply that the travel time on such a roadway segment depends on the departure time from the upstream node. For instance, many countries are practicing curfews, e.g., time constraints for hazardous materials shipments on certain segments of the roadway network (Cox and Turnquist, 1986) in order to protect any local sensitive facilities (e.g. tunnels) from exposition to the associated risk. This type of practices implies that parts of the network are available only within specific time windows, thus strengthening the assumption on the dynamic nature of travel times. Recent studies claim that risk attributes like the population exposure and the accident probability may vary considerably for different parts of a day. For instance, it has been observed that the accident probabilities are higher at night. Moreover, the expected population exposure at risk also varies with time since the population density at urban areas changes with the daily activities and the associated mobility of the residents (Erkut and Alp, 2007). Assuming time-dependent travel time and risk values on each arc of the network, the proposed problem is formulated as a bicriterion shortest time-dependent path problem with fixed mandatory intermediate stops constrained within specified service time windows. It should be clarified that by assuming time-dependent cost (risk and travel time) attributes for the arcs of the underlying network, any solution to the above routing problem cannot be solely identified by a path. Any changes in the schedule of traversing a path might lead to different cost values. Thus, any solution to the above path-finding problem involves the path enhanced with the corresponding departure time from each node included in the path. Each solution of this type will be referred to as a scheduled path. The proposed bicriterion routing and scheduling problem is NP-Complete since it constitutes a generalisation of the classic bicriterion shortest path problem which has been proved to be NP-Complete (Garey and Johnson, 1979; Warburton, 1987). The computational complexity of the proposed problem constitutes an impediment for solving large instances (on real life roadway networks) of the problem. Alternatively, in the decision making context under study, it might be sufficient to determine and assess a set of alternative scheduled paths with marginal deviation from the optimal solution (in terms of travel time). In this perspective, a decision maker could envisage to investigate alternative near optimal solutions with improved risk performance thus capturing the trade-off between risk and travel time in the neighbourhood of the travel time optimal solution. A solution approach has been developed aiming at identifying the k-shortest scheduled paths. The objective of this approach is not to identify the entire set of non-dominated solutions but only a subset of them lying closer to the minimum travel time solution. The proposed solution algorithm extends the work presented in (Androutsopoulos and Zografos, 2008) which deals with the elementary (i.e., without intermediate stops) k-shortest time-dependent path problem. The new algorithm is designed so as to deal with multiple mandatory intermediate stops and discard any equivalent solutions, i.e., scheduled paths sharing the same sequence of nodes, common departure time from the origin, travel time and risk value but different schedule. The proximity of the k-shortest time-dependent paths with the corresponding non-dominated solutions is assessed for a set of problems developed by the authors. A label setting algorithm based on the Decrease Order Time technique (Chabini, 1998) has been developed for identifying the entire set of non-dominated and non-equivalent solutions to each test problem. The remainder of this paper consists of six sections. Sesction 2 presents an overview of the literature on hazardous materials routing and scheduling while Section 3 provides the formulation of the proposed problem. Section 4 provides the solution approach for identifying the k-shortest solutions while Section 5 presents the proposed exact solution approach. Both solution approaches have been applied on a set of problems developed by the authors, and their results are presented in Section 6. Finally, Section 7 provides the concluding remarks arising from the work presented in this paper and indicates directions for future research.

2. Previous related work In the case of non-negative, time-invariant and deterministic arc cost attributes, the relevant bicriterion path-finding problem with mandatory intermediate stops can be addressed by solving each of the arising intermediate bicriterion short-

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est paths problem with any of the relevant existing label setting algorithms (Climaco and Martins, 1982; Hansen, 1980). However, a similar approach cannot be adopted for solving the proposed problem in the case of time-dependent cost attributes. In general the Bellman’s principle of optimality for multi-criteria shortest path problems cannot be transferred for the case of time-dependent cost attributes. For instance, if a scheduled path from i to j through q departing at time si is non-dominated, then it is not necessary that the part of the path from i to q departing at si is non-dominated as well. Hamacher et al. (2006) provide an illustrative example to prove the above statement. Thus, the proposed path-finding problem cannot be addressed by simply solving the intermediate bicriterion time-dependent shortest path problems between any pair of subsequent stops. To the knowledge of the authors no research effort has been focused on dealing with this problem. In the existing literature, two major categories of dynamic path-finding problems for the hazardous materials transportation are encountered: (i) dynamic and deterministic and (ii) dynamic and stochastic. In the dynamic and deterministic path-finding problems, the arc travel time and/or any risk attributes are assumed time-dependent, i.e., the travel time and risk on any arc are expressed as functions of the departure time from the upstream node. Assuming time-dependent arc travel times, Nozick et al. (1997) formulated the integrated routing and scheduling problem for hazardous materials transportation as a multi-objective shortest path problem in a network with time-dependent cost attributes. In their study, they adapt the forward label setting algorithm proposed by Cox (1984) for the time-invariant multi-objective shortest path problem, in order to solve the emerging multi-objective routing and scheduling problem. Their algorithmic approach is heuristic since part of the non-dominated solutions may be omitted. In particular, the Cox’s algorithm is based on the adaptation of the Bellman’s forward optimality principle for the multi-objective shortest path problem, which, as mentioned above, does not hold for a network with time-dependent arc cost attributes (Hamacher et al., 2006; Nozick et al., 1997). Erkut and Alp (2007) enhanced the above integrated routing and scheduling problem by permitting en-route stops. They formulated the emerging routing problem as a constrained shortest path problem in a network with time-dependent arc travel time and risk values. The objective of their formulation was to determine the minimum risk path under the constraint that the total travel time of the path does not exceed a specified threshold. The solution algorithm they propose for the constrained time-dependent shortest path problem is based on the dynamic programming technique introduced in Cai et al. (1997) which utilises the adaptation of the Bellman’s backward optimality principle for the multi-criteria shortest timedependent problem. On the other hand, the dynamic and stochastic path-finding problems involve arc travel time and/or risk attributes which are assumed random variables with time-dependent probability distribution. Along this line, Hall (1986) modeled the arc travel time as a discrete random variable with time-dependent probability distribution function and presented a branch and bound algorithm for solving the least expected time path problem. In the same context, Miller-Hooks and Mahmassani (1998a,b) adapted the label correcting algorithm for time-dependent shortest path problems by Ziliaskopoulos and Mahmassani (1993) for determining the least possible time path and a lower bound of the probability of occurrence of the least possible travel time. Miller-Hooks and Mahmassani (2000) developed a label correcting algorithm for determining the least expected time path, based on the assumption that the arc travel time random variables are independent. Miller-Hooks and Mahmassani (1998a,b) propose label correcting procedures for determining pareto-optimal paths in a network with stochastic, time varying link cost attributes, by utilising three alternative stochastic dominance criteria. Chang et al. (2005) assumed normal probability distribution functions for the cost attributes on each arc. Under this assumption Chang et al. developed procedures for propagating means and variances along paths and incorporated them in a best first search (Ahuja et al., 1993) solution algorithm for solving the multi-objective dynamic and stochastic shortest path problem. Finally, an additional line of research involves independent stochastic time-invariant arc cost attributes. Wijeratne et al. (1993) formulate the hazardous materials routing problem from a single origin to a single destination as a stochastic multiobjective shortest path problem. They introduce two criteria of stochastic dominance for comparing the probability distributions of any pair of alternative paths: (i) comparing the mean value and variance of the paths and (ii) comparing the range of the attribute values for which the cumulative probability function of one path overruns the corresponding function of the other. The authors incorporate the above stochastic dominance criteria in the label setting algorithm for bicriteria shortest path problems proposed by Cox (1984). A major finding from the relevant studies on multi-criteria path-finding problems implies that assuming dynamic and stochastic arc travel time and risk values leads to a broader set of non-dominated solutions than in the case of dynamic and deterministic values. On the other hand, this last observation implies that a heavy computational burden and immense storage facilities would be required identifying the entire set of non-dominated solutions. The work in this paper is based on the assumption that the arc travel time and risk values are deterministic and time-dependent while a set of prespecified mandatory intermediate stops with arrival time windows are involved between the origin and the destination. In particular, the work in this paper extends the work in Nozick et al. (1997) and Erkut and Alp (2007) by assuming intermediate mandatory stops and arrival time windows. In the present work the solutions are further refined by excluding equivalent paths, i.e. solutions with identical paths, departure time, total travel time and risk, which only differ in the waiting time at the nodes.

3. Definition of the problem Assume that graph G(N, A) denotes a roadway network, where N is the set of nodes, i.e., roadway junctions, origin, destination, and potential intermediate stops, and A the corresponding set of arcs. Each arc (i, j) of the graph is associated

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with two non-negative cost attributes, namely travel time and risk. Both attributes are assumed time-dependent, i.e., their value depends on the departure time (s) from the upstream node (i), denoted by cs1 ði; jÞ and cs2 ði; jÞ respectively. In graphs with time-dependent cost attributes, as defined above, the classic definition of a path is not sufficient to express the solutions of any shortest path problem. It is evident that traversing the same path for different departure times from the origin or any intermediate node may lead to different total travel time or/and risk value. Thus, it is essential to extent the classic definition of a path as in Definition 1 below, where the time of leaving each node of a path is also included. Definition 1. Any path in G(N, A) between two nodes i0 and in enhanced with the departure time (sdiq ) from any constituent node iq of the path, is called scheduled path and it is denoted by ps[i0, in] where s is the earliest possible departure time (ready time) from the origin i0. Any scheduled path may be written as in (1):

ps ½i0 ; in  :¼

  sd sd sd sd ði0 ; i1 Þ i0 ; . . . ; ðiq1 ; iq Þ iq1 ; ðiq ; iqþ1 Þ iq ; . . . ðin1 ; in Þ in1

ð1Þ

sd

where ðiq ; iqþ1 Þ iq denotes the arc (iq, iq+1) of the path ps[i0, in] traversed at time sdiq . The departure time (sdiq ) from any constituent node iq of a scheduled path that includes arc (iq, iq+1), takes values on a discrete time horizon [0, T] and satisfies constraint (2). sd

sd

sdiq þ c1iq ðiq ; iqþ1 Þ 6 sdiqþ1 ; ðiq ; iqþ1 Þ iq 2 ps ½i0 ; in 

ð2Þ

Inequality (2) implies that the departure from any node (apart from the destination) of a scheduled path should occur following to the arrival at it. The arrival time (saiq ) at any node iq of a scheduled path is defined by (3). sdi

sd

saiq :¼ sdiq1 þ c1 q1 ðiq1 ; iq Þ; ðiq1 ; iq Þ iq1 2 ps ½i0 ; in 

ð3Þ

The path-finding problem presented in this paper involves an origin s0 e N, a destination sn+1 (not necessarily different from s0) and a sequence of n intermediate stops of a given route R located at the nodes in R ¼ fs1 ; . . . ; sn g # N. The departure e l from the origin is constrained within a time window ½ds0 ; ds0  while the arrival at the destination is also constrained within ½aesnþ1 ; alsnþ1 . Unbounded waiting is assumed on any node of the network, in the sense that each node of the network is associated with a rest area where the truck could temporarily park. However no waiting (or bounded waiting time) might apply for the intermediate stops, i.e. the truck is not allowed to wait at the premises of the relevant customer, although unbounded waiting time might be allowed in the neighbourhood expressed through the relevant node of the network. In order to resolve this issue, the underlying network is slightly modified by creating a dummy node s0q for any node hosting an intermediate stop sq. The set A is enhanced with the dummy arcs (sq ; s0q ) and (s0q ; sq ) each one having zero travel time and risk. In this context, the stops are assumed located on the new dummy nodes, while the original nodes are treated as any other node of the network. Thus, unbounded waiting time is allowed on the original nodes while waiting at an intermediate stop s0q is allowed only until time alsq is reached. However, in the remaining of the paper the notation used for the intermediate stop remains the same, i.e. sq. Each intermediate stop (sq) is associated with a service time (t sq ) and a given service time window denoted by ½aesq ; alsq , where aesq denotes the earliest time that service at stop sq may start and alsq denotes the corresponding latest service start time. A truck is not allowed to arrive later than alsq , while if it arrives earlier than aesq then the service of the relevant customer has to wait until time aesq is reached. This is a common feature in freight distribution problems where each customer requests from the carrier to arrive at the delivery location within a specified time window. Moreover, the service time window for any e l intermediate stop sq imposes implicitly a departure time window ½dsq ; dsq  as defined in (4) and (5) e

ð4Þ

l

ð5Þ

dsq :¼ aesq þ tsq dsq :¼ alsq þ tsq

Formulae (4) implies that the departure from the stop sq could not occur earlier than the earliest possible completion time of the service at that stop. Similarly, formulae (5) implies that the departure from sq could not occur later than the latest possible completion time of the service at that stop. The objective of the proposed bicriterion routing and scheduling problem involves the determination of the scheduled path from the origin s0 to the destination sn+1 passing through the mandatory intermediate stops in R respecting the service time windows and minimising the total travel time and the total risk. The above special type of scheduled paths are named scheduled route-paths (since they traverse the entire truck route), and they are denoted as in (6).

ps ½s0 ; R; snþ1  ¼ fps0 ½s0 ; s1 ; ps1 ½s1 ; s2 ; . . . ; psn1 ½sn1 ; sn ; psn ½sn ; snþ1 g where each ps[sq, sq+1] denotes an intermediate scheduled path.

ð6Þ

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The total travel time and risk metrics calculated for each alternative scheduled route-path are given by (7) and (8).

c1 ðps0 ½s0 ; R; snþ1  :¼

X

ðcs1 ði; jÞ þ t w i Þ

ð7Þ

ði;jÞs 2p

c2 ðps0 ½s0 ; R; snþ1 Þ ¼

X

psij C sij

ð8Þ

ði;jÞs 2p

s where p  ps0 ½s0 ; R; snþ1 , t w i is the waiting time at node i, pij is the probability of accident on arc (i, j) traversed at time s, and C sij denotes the measure of expected consequences arising in case of an accident occurring at arc (i, j) at time s. The metric (7) expresses the total en-route duration of the relevant sequence of the constituent scheduled paths including the waiting at the intermediate stops. Concerning the risk metric, several measures are confronted in the relevant literature, e.g., population exposure (Batta and Chiu, 1988), incident probability (Saccomanno and Chan, 1985), perceived risk (Abkowitz et al., 1990), and expected disutility (Erkut and Ingolfsson, 2000). The present research has adopted the perceived risk measure given by (8). A major feature of the above bicriterion time-dependent shortest path problem is that it aims to determine the set of nondominated scheduled route-paths based on the following definition: ps1 ðs0 ; R; snþ1 Þ is non-dominated for time s if and only if there does not exist a scheduled route-path ps2 ðs0 ; R; snþ1 Þ; ps2 ðs0 ; R; snþ1 Þ – ps1 ðs0 ; R; snþ1 Þ with cq ðps1 ðs0 ; R; snþ1 ÞÞ P cq ðps2 ðs0 ; R; snþ1 ÞÞ8q and cq1 ðps1 ðs0 ; R; snþ1 ÞÞ > cq1 ðps2 ðs0 ; R; snþ1 ÞÞ for at least one q1 2 f1; 2g. Note that any two scheduled route-paths solving the above problem which depart from the origin at the same time, share the same path (i.e. sequence of nodes), and have equal total travel time and risk, they are non-dominated. However, identifying this type of non-dominated solutions does not provide any added value in the specific decision making process, since the only difference among them pertains to the waiting times at the nodes of their (common) path. The above type of solutions is defined below.

Definition 2. Any two scheduled paths ps1 ði0 ; in Þ and ps2 ði0 ; in Þ which share a common path from i0 to in departing from node i0 at time s and c1 ðps1 ½i0 ; in Þ ¼ c1 ðps2 ½i0 ; in Þ; c2 ðps1 ½i0 ; in Þ ¼ c2 ðps2 ½i0 ; in Þ, they are called equivalent. Any two scheduled routepaths ps1 ½i0 ; R; in  ps2 ½i0 ; R; in  are non-equivalent if there is atleast one pair of constituent scheduled paths ps1 ½isq ; isqþ1  and ps2 ½isq ; isqþ1 , which are non-equivalent as well. In this context, this paper deals with the determination of the non-equivalent solutions of the proposed bicriterion routing and scheduling problem. 4. Determining the k-shortest non-equivalent scheduled route-paths The proposed solution approach starts from the destination and traverses backwards the stops of the route iteratively, calculating the k-shortest scheduled paths from an intermediate stop sq to the destination sn+1 through the stops {sq+1, . . . , sn} for any possible departure time. Thus, in essence the k-shortest scheduled route-path problem is decomposed to a series of nested k-shortest scheduled path problems. A label setting algorithm has been developed in order solve each of the above problems. The proposed label setting algorithm is based on the work presented in (Androutsopoulos and Zografos, 2008) where the elementary k-shortest time-dependent path problem is addressed under the assumption of unbounded waiting times at the nodes of the network. This algorithm is further enhanced for identifying only the kshortest non-equivalent solutions. A theoretical discussion is included in the remainder of this section in order to justify this enhancement. 4.1. Previous related work on the k-shortest path problem In general, the k-shortest path problem arises in many routing decisions involving the determination of more than one alternative paths. The objective of the classic k-shortest path problem is to determine the first k-minimum cost paths in a network where each arc is associated with a static cost attribute. Depending on whether cycles are allowed or not in the paths, two different versions of the problem arise. The research studies by Eppstein (1998) and Martins (1984) were focused on the k-shortest looping path problem. On the other hand, several studies have been focused on the k-shortest loopless paths including Yen (1971), Katoh et al. (1982), and Hadjiconstantinou and Christofides (1999). This paper studies an extension of the classic k-shortest path problem involving the determination of k-shortest (loopless) non-equivalent scheduled paths on a network with time-dependent travel times. Although substantial research effort has been devoted to the time-dependent shortest path problem (Cooke and Halsey, 1966; Ziliaskopoulos and Mahmassani, 1993; Cai et al., 1997; Chabini, 1998; Ahuja et al., 2003) limited research has been found in the literature addressing the k-shortest time-dependent path problem. Miller-Hooks and Mahmassani (1998a,b) have studied the k-shortest path problem in a network with time-dependent arc travel times. They propose a dynamic programming algorithm founded on the label correcting technique of Ziliaskopoulos and Mahmassani (1993) for determining at most k-shortest paths. Their approach is heuristic in the sense that eventually only j(
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particular, Chen et al. (2001) studied the k-minimum cost paths problem in a time-schedule network. The objective of their formulation is to determine k-minimum cost paths with total travel time less than a threshold value s. The corresponding network involves arcs with time-invariant travel times and non-negative cost attributes while the departure from any node is allowed only at specified discrete points in time. The solution algorithm proposed by Chen, Rinks and Tang involves the creation of the time expanded version of the network and the implementation of the Yen’s algorithm for the classic k-shortest path problem. Along this line, Chen and Tang (1998) proposed an algorithm for solving a variation of the above problem where the departure time from any node is allowed only within specified time windows. In particular, when an arrival occurs at a node of the relevant network, two options are possible: either leave the node immediately if the arrival lies within a given time window or otherwise wait until the next time window is reached. The solution algorithm for this problem follows the structure of Yen’s algorithm where instead of determining simple shortest paths, the entire set of alternative scheduled paths with the same total travel time but different waiting times are identified. It is evident that the abovementioned k-shortest path problems constitute special cases of the problem studied in this paper which involves time-dependent (instead of time-invariant) cost attributes, multiple intermediate mandatory stops, and time windows on the origin, the destination, and the intermediate stops. 4.2. Solution approach The proposed algorithmic approach consists of n + 1 stages indexed by v 2 Q  f0; . . . ; ng. At each stage v 2 Q , the kshortest (non-equivalent) scheduled paths from any node of the network to the destination sn+1 through the v intermediate e l e stops from snv+1 to sn for any possible departure time s, i.e., s 2 ½dsnv ; dsnv  for the node snv and s 2 ½dsnv ; alsnv þ1  for any other node. Given the k-shortest scheduled paths from snv to sn, the proposed algorithm proceeds to stage (v + 1) where the k-shortest scheduled paths from snv1 to sn through {snv, snv+1,. . .,sn} are determined. Each of the scheduled paths determined in stage (v + 1) consists of: (i) a sub-path from the origin to the intermediate stop snv and (ii) any of the k-shortest scheduled paths from snv to sn (through {snv+1, . . ., sn}) determined in stage (v). The algorithm that solves each of the above mentioned nested k-shortest non-equivalent scheduled path problems is based on the algorithm proposed in (Androutsopoulos and Zografos, 2008) for the elementary k-shortest time-dependent path problem. The proposed algorithm enhances the one presented in (Androutsopoulos and Zografos, 2008) on the basis of identifying only non-equivalent solutions and treating multiple intermediate stops between the origin and the destination. It is proved that under the assumption of unbounded waiting, the exclusion of any equivalent scheduled paths does not pose a risk in omitting any of the k-shortest non-equivalent scheduled paths. The intuition underlying the proposed algorithm is founded on the Proposition 1 presented below. Definitions 3 and 4 presented below are essential for stating and proving Proposition 1. Definition 3. Assume a scheduled path ps1 ½i0 ; in  :¼ fði0 ; in Þs0 ; . . . ; ðiq1 ; iq Þsq1 ; ðiq ; iqþ1 Þsq ; . . . ; ðin1 ; in Þsn1 g. Any scheduled path s ps20 ½iq ; in  :¼ fðiq ; iqþ1 Þsq ; . . . ; ðin1 ; in Þsn1 g with sq1 þ c1q1 ðiq1 ; iq Þ 6 s0 6 sq is called scheduled sub-path of ps1 ði0 ; in Þ. Definition 4. Any two scheduled paths psr1 ½i0 ; in  and psr2 ½i0 ; in  for which c1 ðpsr1 ½i0 ; in Þ ¼ c1 ðpsr2 ½i0 ; in Þ, they are called timeequivalent solutions. Proposition 1. If a scheduled path ps1 ½i0 ; in  is one of the k-shortest loopless scheduled paths from i0 to in with ready time s, then its sub-path ps20 ½iq ; in  is either one of the k-shortest solutions from iq to in at time s0 or it is time-equivalent with one of them. Proof. It will be proved by contradiction. Assume that there exist at least k scheduled paths psðrÞ0 ½iq ; in  for which sq1 s0 ½i ; i Þ for any r 2 f1; . . . ; kg . Joining the first part of ps ði ; i Þ, i.e., fði ; i Þs0 ; . . . ; ði g, with each c1 ðps20 ½iq ; in Þ > c1 ðpðrÞ q n 0 n q1 ; iq Þ 1 0 n of psðrÞ0 ðiq ; in Þ creates k new scheduled paths. If the new scheduled paths created as above contain no loop then they have strictly lower total travel time than ps1 ði0 ; in Þ, leading to a contradiction with the hypothesis of this proposition. If on the other hand any of the k new scheduled paths contains a cycle, then by replacing the cycle with waiting time at the node revisited, leads to a new scheduled path with no cycle and equal travel time.Thus, in either case the findings contradict with the hypothesis that ps1 ½i0 ; in  is one of the k-shortest scheduled paths. h Proposition 1 implies that if a sub-path of ps1 ½i0 ; in  is not within the k-shortest scheduled paths, then it will be time-equivalent with the corresponding kth shortest scheduled path. A label setting algorithm is proposed for solving the corresponding k-shortest nested scheduled path problem arising at each stage

v 2 Q.

s;½m

s;½m

Each node of the network is initially associated with a list of k vector-labels fðkðrÞ;c1 ðiÞ; kðrÞ;c2 ðiÞÞ :

r ¼ 1; . . . ; kg for every point in time

s2

e ½dsnv ; alsnv þ1 

and stage

v

s;½v 

of the problem, denoted by Ki

s;½v 

kðrÞ;c1 ðiÞ of each vector-label denotes the total travel time of the rth shortest scheduled path

ps

. The first attribute

ðrÞ ½i; fsnv þ1 ; . . . ; sn g; snþ1 

which

departs from node i no earlier than time s, while the second attribute of the vector-label denotes the risk value of the cors;½v 

responding scheduled path. Each vector-label is associated with a set F ðrÞ ðiÞ which includes the nodes visited by the corres;½v 

sponding scheduled path. In addition, a pointer wðrÞ ðiÞ is attached to each vector-label, indicating the next vector-label of the

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corresponding rth shortest scheduled path. The role of this pointer is on facilitating the backtracking of each of the k-shortest scheduled paths from any node i to the destination.

Fig. 1. Steps of the algorithm for solving the k-shortest scheduled path problem.

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l

For the stage v: = 0, where the k-shortest scheduled path problem from sn to sn+1 for any departure time s 2 ½dsn ; dsn  is solved, the proposed algorithm initializes the vector-labels associated with the destination (sn+1) by setting both attributes equal to zero. The attributes of the vector-labels associated with any other node are set equal to a very large number M. For any other stage v – 0 the travel time and risk attributes of the initial vector-labels associated with the node snv+1 for time s 2 ½aesnv þ1 ; alsnvþ1  are set equal to the value of travel time and the risk value of the corresponding vector-labels for stage e (v  1) at time s0 :¼ s þ tsnv þ1 . A vector-label is also created for the node snv þ1 for any time dsnv 6 s 6 aesnv þ1 by setting the travel time equal to the corresponding travel time of the earliest vector-label from stage (v  1) plus the service time tsnv þ1 plus the waiting time (aesnv þ1  s), and the risk equal to the corresponding risk value of the earliest vector-label from stage (v  1). e Following to the vector-labels initialisation, each stage (v) of the algorithm involves Tð:¼ alsnv þ1  dsnv þ 1Þ iterations e l moving backwards in time, starting from time s :¼ asnv þ1 down to dsnv . For each iteration s the k-shortest scheduled paths from any node i to destination node sn+1 through the v intermediate stops {snv+1, . . . s, sn} departing from i at time s are determined. Each iteration s involves the identification of a set of candidate vector-labels for every node i – snv+1. Given any arc (i, j) e A (outgoing from node i) and K(P k) vector-labels already identified for the node j at time s0 ð:¼ s þ cs1 ði; jÞÞ, then K candidate vector-labels are identified for node i, referring to the corresponding scheduled paths emerging from joining the arc (i; j) with the scheduled paths associated with the existing vector-labels of node j at time s0 , where s0 is the arrival s;½v  time through arc ði; jÞ at node j. The above procedure is repeated for every arc outgoing from node (i). Thus, if i R F ðrÞ ðjÞ (in s0;½v  s0;½v  s0;½v  a new candidate label (usc1 ðiÞ, usc2 ðiÞ) is calculated order to avoid looping paths), for each vector-label (kðrÞc1 ðjÞ,kðrÞc2 ðjÞ) 2 Kj s0;½v 

s0;½v 

for node i by adding the travel time and risk of ði; jÞs to kðrÞc1 ðjÞ and kðrÞc2 ðjÞ, respectively. It should be clarified that the search for a loop is performed for the part of the corresponding scheduled path between node i and snv þ1 . Thus, node i may be revisited by the part of the scheduled path after the intermediate stop snv þ1 . An exemption to the above scanning process arises when i ¼ snv for which no scanning is performed for any s R ½desnv ; dlsnv , since any such solution contravenes the relevant departure time window constraint. This new candidate vector-label is checked if it is one of the k shortest labels for time s by comparing it with the existing vector-labels (if any). If the scheduled path associated to the new candidate vector-label is time-equivalent with the existing kth scheduled path (i.e. ;½v  s;½v  s;½v  ucs1;½v  ¼ ksðkÞc ðiÞ) then the vector-label is also included in the list of vector-labels Ki . Thus, any set Ki may include more 1 than k vector-labels. However, if the scheduled path associated to a candidate vector-label is equivalent with any of the s;½v  existing scheduled paths associated to the labels in Ki , then the corresponding candidate vector-label is excluded from further consideration. Upon termination of the iterations, the lists of k (or more) vector-labels determined for the stop snv for each point in time s 2 ½desnv ; dlsnv  correspond to the k-shortest scheduled paths. Moreover, upon termination of the entire process (n þ 1 stages), the k-first vector-labels in Kss0;½nþ1 refer to the k-shortest scheduled paths. If more than k labels are included in Kss0;½nþ1 , then kshortest vector-labels are selected. The proposed algorithm is provided in the form of pseudocode in Fig. 1. Lemma 1. Assume that k loopless scheduled paths from node i1 to in departing at time s are non-equivalent. If the k scheduled paths emerging from adding ði0 ; i1 Þs0 in front of each of the above scheduled paths arriving at i1 earlier than s are loopless then they are non-equivalent as well. Proof. Assume any k loopless non-equivalent scheduled paths psr ði1 ; in Þ for r ¼ 1; . . . k, where psr ði1 ; in Þ ¼ fði1 ; ir1 Þs ; . . . ; ðirm ; in Þsrm g. Since the k scheduled paths are non-equivalent, then from the Definition 2 the following statement holds for any pair of them: (i) if they involve the same sequence of nodes then they differ in at least one of the total travel time or the risk and (ii) if they have the same travel time and risk value then each of them includes different sequence of nodes. Based on the above statements, any pair of the arising scheduled paths either differ in at least one cost attribute (if the corresponding initial scheduled paths also differ in at least one attribute) or they have a different sequence of (if the initial scheduled paths do so as well). Thus, in any case the new k scheduled paths are non-equivalent as well. h In the above proof, If any of the emerging k scheduled paths contain a loop, then the scheduled path arising from replacing the loop with a waiting time, might be equivalent with any of the remaining k  1 scheduled paths. Lemma 2. If two scheduled paths ps1 ði1 ; in Þ and ps2 ði1 ; in Þ are equivalent, then the corresponding scheduled paths emerging from 0 adding ði0 ; i1 Þs in front of each of them are equivalent as well. Proof. It derives directly from Definition 2. h Remark 1. In the proof of Lemma 1, if any of the k new scheduled paths psr ði0 ; in Þ includes a loop, then the scheduled path arising from replacing the loop with waiting time, might be equivalent with any of the remaining k  1 scheduled paths. Thus in the context of the algorithm, when new scheduled paths are created by scanning a node, one cannot exclude the case that the new loopless scheduled path may be equivalent with any of the scheduled paths created by adding one unit of waiting time at that node. This is the reason why it is imperative for check of non-equivalent paths throughout the iterations of the proposed algorithm. Moreover, based on Lemma 2, by excluding any solution equivalent with any of the already existing solutions does not include any risk of excluding in advance any of the k-shortest non-equivalent solutions.

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Proposition 2 (Proof of correctness). The vector-labels solutions determined by the proposed algorithm are associated with the k-shortest non-equivalent scheduled paths. Proof. Based on the steps of the algorithm any scheduled path including a cycle or being equivalent with any of the existing solutions is excluded from further consideration. Actually, any of the scheduled paths that contains a cycle and its value is included within the k minimum, it is implicitly replaced with a similar scheduled path which instead of having the loop, it involves waiting time at the node revisited equal to the duration of the corresponding loop. Therefore, it is sufficient to prove that at any stage v of the algorithm, the k-shortest scheduled paths from every node i to the destination snþ1 through e {snv þ1 ; . . . ; sn } for any time s 2 ½dsnv ; alsnv þ1  are determined. It can be proved by induction. For stage v :¼ 0, it is evident that at time s ¼ alsnþ1 the vector-labels determined for any node are unique and therefore optimal. Assume that from time s ¼ alsnþ1 s;½0 down to time s0 þ 1 the vector-labels identified in Kiq , for each node iq and any time s such that s0 þ 1 6 s 6 alsnþ1 , are the kshortest. Then it is sufficient to prove that the k vector-labels determined by the algorithm for any node iq at time s0 are indeed the k-shortest for departure time s0 . This last statement will be proved by contradiction. Assume that there exists sd a loopless scheduled path ps0 ½iq ; snþ1  which consists of the following sequence of arcs {ðiq ; iqþ1 Þ iq , ðiqþ1 ; iqþ2 Þ

sdi

qþ1

;. . .,ðiqþm ; snþ1 Þ

sdi

qþm

} and it is one of the k-shortest non-equivalent paths not identified by the proposed algorithm, s ;½0

s ;½0

s ;½0

0 0 ðiq Þ; kðrÞc ðiq ÞÞr ¼ 1; . . . ; jKiq0 i.e., there does not exist a label ðkðrÞc 1 2

j associated with it, while in addition it is non-equivalent with any of k-shortest scheduled paths with determined by the algorithm. A necessary condition for ps0 ½iq ; sn  being within s ;½0

s ;½0

the k-shortest paths while not associated with any label in Kiq0 Based on Proposition 1, the sub-path p

sai

qþ1

½iqþ1 ; snþ1  :¼ fðiqþ1 ; iqþ2 Þ

sdi

s ;½0

0 is that 9r : c1 ðps0 ½iq ; snþ1 Þ 6 kðrÞc ðiq Þ; r 2 f1; . . . ; jKiq0 1

qþ1

; . . . ðiqþm ; snþ1 Þ

sdi

qþm

jg.

} of ps0 ½iq ; snþ1  is either among the sai

½iqþ1 ; snþ1  already deterk-shortest for node iqþ1 at time saiqþ1 or it is time-equivalent with one of them for node q þ 1, i.e., pðhqþ1 1Þ mined by the algorithm (based on our hypothesis that the algorithm determines the k-shortest vector-labels for any node and s P s0 þ 1). Thus based on the steps of the algorithm, a vector-label for path ps0 ðiq ; snþ1 Þ should have been determined for node iqþ1 for time s, leading to a contradiction with the hypothesis that the scheduled path ps0 ðiq ; snþ1 Þ is not represented s ;½0 in Kiq0 . Assume that the algorithm identifies the k-shortest scheduled paths for v ¼ v 0 . Then by following the same procedure as above it can be proved that the algorithm determines the k-shortest scheduled paths for the stage v ¼ v 0 þ 1 , thus proving the proposition. h

5. Determining the set of non-dominated non-equivalent solutions A label setting algorithm has been developed for identifying the entire set of non-dominated non-equivalent solutions, based on the adaptation of the Bellman’s backward optimality principle (Hamacher et al., 2006) for networks with multiple time-dependent non-negative cost attributes. The adapted Bellman’s optimality principle implies that if a scheduled path from any node i departing at si to node j passing through node q at time sq is non-dominated for time si then the sub-path from q departing at time sq is non-dominated for time sq . The proposed algorithmic approach is based on decomposing the bicriterion scheduled path-finding problem with mandatory intermediate stops to a series of bicriterion nested shortest scheduled path problems starting from the destination and traversing the route backwards. Solving each of the ðn þ 1Þ bicriterion nested shortest scheduled path problems constitutes a separate stage of the algorithmic approach. In particular, at each stage v the non-dominated solutions from snv to snþ1 through stops {snv þ1 ; . . . ; sn } are calculated by enhancing (backwards) the non-dominated non-equivalent scheduled e l paths from snv þ1 to snþ1 through the stops {snv þ2 ; . . . ; sn } for each possible departure time s 2 ½dsnv ; dsnv . The algorithmic procedure that solves each of the bicriterion nested shortest scheduled path problems is based on the adaptation of the Decreased Order Time (DOT) technique (Chabini, 1998) for the bicriterion case. For the purposes of the proposed label setting algorithm, each node i of the network is associated with the vector-labels s;½v  s;½v  s;½v  s;½v  ;½v  ðiÞ, nw;c ðiÞ), the pointer ww ðiÞ, and the set F sw;½v  ðiÞ, where nw;c ðiÞ and nsw;c ðiÞ denote the total travel time and risk (respec(nw;c 1 2 1 2 tively) of the wth non-dominated scheduled path from node i to node snþ1 through the stops {snv þ1 ; . . . ; sn } at the vth stage. s;½v  ðiÞ denotes the list The pointer wsw;½v  ðiÞ points to the next vector-label in the corresponding scheduled path, while the set F w of nodes already visited by the above mentioned wth scheduled path. Keeping track of the visited nodes of a scheduled path facilitates the check for tracing and excluding equivalent solutions. In addition, a set Ns;½v  ðiÞ is created for each node i which includes the corresponding non-dominated vector-labels for node i at time s and stage v . Fig. 2 presents the steps of the proposed algorithm for any stage v 2 Q :¼ f0; . . . ; ng. Note that upon the completion of stage n, the entire set of non-dominated non-equivalent solutions are identified for each possible departure time from the origin s0 . The operator  denotes the inclusion of a set of candidate vector-labels in a set of non-dominated non-equivalent vectorlabels. This operator implies that each candidate vector-label is assessed in terms of dominating or being dominated by each element of the set of currently non-dominated vector-labels. If the candidate vector-label is dominated by at least one element of this set then it is disregarded. In case that it dominates any of the elements of the set of non-dominated vector-labels, then the corresponding element is excluded from the set. If by the end of this process the candidate vector-label is not found either equivalent with or dominated by any of the elements of the set of the temporarily non-dominated vector-labels, it also becomes an element of this set.

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Fig. 2. Steps of the algorithm for solving the bicriterion scheduled path problem.

On termination of the proposed algorithm, each set Ns;½v þ1 ðs0 Þ includes the vector-labels referring to the entire set of nondominated non-equivalent scheduled route-paths from s0 to snþ1 through {s1 ; . . . ; sn } for each possible departure time s: Lemma 3. If Ns;½v  ðsnv þ1 Þ includes the vectors-labels associated to a set of non-dominated non-equivalent solutions for time s, then the corresponding vector-labels in Nstsnv þ1 ;½v þ1 ðsnv þ1 Þ emerging through the proposed algorithm, relate to non-dominated non-equivalent scheduled paths as well. Proof. The vector-labels in Nstsnv þ1 ;½v þ1 ðsnv þ1 Þ correspond to non-equivalent scheduled paths since the only difference with the vector-labels in Ns;½v  ðsnv þ1 Þrelates to the departure time. Thus, it is sufficient to prove that the vector-labels are s0;½v þ1 s0;½v þ1 non-dominated. For any vector-label (nw;c ðsnv þ1 Þ, nw;c ðsnv þ1 ÞÞ 2 Nstsnv þ1 ;½v þ1 ðsnv þ1 Þ there exist a vector-label 1 2 ;½v  (nsw;c ðsnv þ1 Þ, 1

s0;½v þ1

nw;c2

;½v  nsw;c ðsnv þ1 ÞÞ 2 Ns;½v  ðsnv þ1 Þ 2

such

that

s0;½v þ1 s0;½v þ1 ;½v  ðnw;c ðsnv þ1 Þ; nw;c ðsnv þ1 ÞÞ :¼ ðnsw;c ðsnv þ1 Þ þ t snv þ1 þ tw ; 1 2 1

w

ðsnv þ1 ÞÞ where t denotes the waiting time at stop snv þ1 . Thus it derives from the above statement that the vector-

s0;½v þ1 s0;½v þ1 ðsnv þ1 Þ, nw;c ðsnv þ1 Þ) are non-dominated as well. h labels (nw;c 1 2

Proposition 3. The proposed algorithm identifies the entire set of non-dominated non-equivalent scheduled paths with intermediates stops for any possible departure time. Proof. I t is sufficient to prove that the proposed algorithm determines the non-dominated non-equivalent solutions for any stage v . This can be proved by induction. For v ¼ 0, the relevant problem simplifies to a bicriterion shortest scheduled path

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problem from sn to snþ1 . Based on the relevant vector-label initialisation process of the algorithm, the destination snþ1 is assoe ciated with a single vector-label for every s 2 ½dsn ; alsnþ1  equal to (0, 0), which is trivially non-dominated. To complete the proof, it is sufficient to show that given the vector-labels in Ns½v  ðsnv þ1 Þ, the initial vector-labels for node snv þ1 in stage v þ 1 are non-dominated and non-equivalent. In particular, assume that the sets Ns;½v  ðsnv þ1 Þ, s 2 ½desnv þ1 ; dlsnv þ1  consist of the non-dominated non-equivalent vector-labels for any possible departure time s from stop snv þ1 to the destination snþ1 through the stops fsnv þ2 ; ::; sn g. Based on Lemma 3, the vector-labels in Ns;½v þ1 ðsnv þ1 Þ calculated by vector-labels initialisation process of the proposed algorithm are non-dominated and non-equivalent. Assume that Ns;½v þ1 ðjÞ 8j and any s P s0  1 for s0 2 f1; . . . ; Tg, contains the entire set of non-dominated non-equivalent vector-labels for time s and node j. It is profound (from the algorithm outline) that the new vector-labels in the sets Ns0 ;½v þ1 ðiÞ are non-dominated and non-equivalent. It suffices to prove that there are no other non-dominated non-equivalent scheduled paths from any node i to node sn+1 through {snv þ1 ; . . . ; sn } departing at time s0. If there was such a scheduled path it would be of the following form: ps0 ½i; fsnv þ1 ; . . . ; sn g; snþ1  :¼ fði; jÞs0 ; ps0 ½j; fsnv þ1 ; . . . ; sn g; snþ1 g, where ps0 ½ j; fsnv þ1 ; . . . ; sn g; snþ1  is non-dominated and nonequivalent (by assumption since s0 P s0  1). However, since ps0 ½ j; fsnv þ1 ; . . . ; sn g; snþ1  is determined by the proposed algorithm, the scheduled path ps0 ½i; fsnv þ1 ; . . . ; sn g; snþ1  should have identified as well through the scanning process of the algorithm. h

6. Computational performance The proposed algorithm for determining the k-shortest non-equivalent scheduled route-path problem has been applied on a set of problems developed by the authors. The major goals of assessing the proposed algorithm were to compare the solutions produced by the algorithm for the k-shortest non-equivalent scheduled route-paths with the corresponding non-dominated solutions identified by the algorithm that solves the bicriterion problem. Three grid-like graphs were developed in this context, i.e., G100 with 100 nodes, G300 with 300 nodes, and G600 with 600 nodes. The in-degree and out-degree of each node of the above graphs is equal to 4, in order to strengthen the networks resemblance to roadway networks. The travel time and risk defined on each arc of the graph were defined on 24 time periods (i.e., with time period length equal to 15 min) starting from 08:00 up to 14:00. The problems generated on graphs G100–300 involved randomly selected origins, destinations, and 1 up to 4 intermediate stops with 1–2 h time windows. In this setting, 100 problems were generated for each combination of graph type and number of intermediate stops. The above problems were solved by both algorithms implemented on a Pentium IV, with Intel processor at 2.99 GHz, and 2 GB RAM. Table 1 presents the average computational time needed for solving the above set of test problems by the algorithm for the k-shortest scheduled route-path problem. Each of these test problems was solved for k equal to 10 and 20. A major finding of the above process is that reasonable computational time was needed for the solution of the test problems ranging from 2 to 72 s. Moreover, Figs. 3 and 4 indicate that the relationship between the average computational time of the proposed algorithm with the number of intermediate stops is linear. Table 1 Average computational time of the algorithm for the test problems. Network id

k

Intermediate stops

Computational time (ms)

G100

10 20 10 20 10 20 10 20

1 1 2 2 3 3 4 4

2355 4591 4197 4185 4404 8996 5749 11 903

G300

10 20 10 20 10 20 10 20

1 1 2 2 3 3 4 4

6665 13 708 10 722 22 625 14 315 30 332 19 489 37 284

G600

10 20 10 20 10 20 10 20

1 1 2 2 3 3 4 4

13 303 27 029 13 302 27 029 30 251 59 175 38 306 71 361

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G100

Computational time (millsec)

G300 35000

G600

30000 25000 20000 15000 10000 5000 0 1-stop

2-stops

3-stops

4-stops

Fig. 3. Average computational time (for k = 10) as a function of the number of intermediate stops.

80000 G100

70000 Computational time (millsec)

G300 G600

60000 50000 40000 30000 20000 10000 0

1-stop

2-stops

3-stops

4-stops

Fig. 4. Average computational time (for k = 20) as a function of the number of intermediate stops.

700 600

Non-dominated 10-shortest

500

Risk

400 300 200 100 0 0

20

40

60

80

100

120

Travel Time

Fig. 5. The non-dominated solutions and the solutions determined by the proposed algorithm (for k = 10) plotted on the travel time vs. risk plane.

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725

700 Non-dominated 20-shortest

600

Risk

500 400 300 200 100 0 0

20

40

60

80

100

120

Travel Time Fig. 6. The non-dominated solutions and the solutions determined by the proposed algorithm (for k = 20) plotted on the travel time vs. risk plane.

Concerning the quality of the solutions provided by the algorithm determining the k-shortest scheduled route-paths, Figs. 5 and 6 below presented graphically the deviation of the k-shortest scheduled route-paths from the corresponding non-dominated solutions for one of the test problems mentioned above. Each of the circles plotted in the diagram (travel time vs. risk) of Fig. 5 denotes a non-dominated (non-equivalent) solution to a test problem defined on network G100 with two intermediate stops. The triangles marked on the same plot diagram denote the solutions that emerged from the algorithm that determines the k-shortest scheduled route-paths, for k equal to 10. Note that the solutions plotted in the diagram are only those that were found non-dominated among the total (k  T) solutions identified by the algorithm. Similarly, Fig. 6 presents the non-dominated solutions (circles) and the corresponding k-shortest scheduled route-path solutions that emerged for k equal to twenty. Figs. 5 and 6 indicate that the solutions determined for the k-shortest scheduled route-path problem are close or coincide with the non-dominated solutions lying close to the area of the minimum travel time solution. Moreover, when k increases a better fit of the two types of solutions (non-dominated vs. k-shortest) emerges.

7. Concluding remarks The bicriterion path-finding problem presented in this paper arises in hazardous materials delivery problems, especially in cases where the sequence of stops per truck is dominated by safety and/or business constraints. In this respect, the sequence of stops is usually specified exogenously, and thus the trade-off between travel time and risk is specified on the basis of the path-finding problem. Moreover, the work presented in this paper could be utilised within any bicriterion vehicle routing and scheduling problem provided that the sequence of stops per route are specified by a separate heuristic algorithm. An algorithm is presented for determining the non-dominated scheduled route-paths. Alternatively a label setting algorithm is proposed for determining the k-shortest non-equivalent scheduled route-paths, thus approximating the non-dominated solutions close to the minimum travel time solution. The computational performance of the proposed algorithm indicates that reasonable computational time is required for solving real life hazardous materials delivery problems. Therefore, the algorithm could be integrated into applications for planning hazardous materials truck routes given the sequence of the relevant stops. The k- shortest scheduled route-paths provided by the proposed algorithm enable the decision maker to perform a comparative assessment of the alternative solutions with respect to their travel time and risk. It should be emphasized however, that the proposed algorithm is accurate only under the assumption that unbounded waiting is allowed at any node of the network. Thus, a significant line for future research arises in exploring the above problem under various constraints on the waiting time at the nodes of the network. Moreover, solving the relevant bicriterion vehicle routing and scheduling problem on time-varying networks constitutes another significant direction for future research.

Acknowledgements The research reported in this paper has been partially supported by the General Secretariat for Research and Technology of the Hellenic Ministry of Development through the Projects SDEF, ENOSIS, and WISETRIP.

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